Lerch zeta function

In mathematics, the Lerch zeta-function, sometimes called the Hurwitz-Lerch zeta-function, is a special function that generalizes the Hurwitz zeta-function and the polylogarithm. It is named after Mathias Lerch [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html] .

Definition

The Lerch zeta-function is given by

:$$L(lambda, alpha, s) = sum_{n=0}^inftyfrac { exp (2pi ilambda n)} {(n+alpha)^s}.

A related function, the Lerch transcendent, is given by

:$$Phi(z, s, alpha) = sum_{n=0}^inftyfrac { z^n} {(n+alpha)^s}.

The two are related, as

:$,$Phi(exp (2pi ilambda), s,alpha)=L(lambda, alpha,s).

Integral representations

An integral representation is given by

:$$Phi(z,s,a)=frac{1}{Gamma(s)}int_{0}^{infty}frac{t^{s-1}e^{-at{1-ze^{-t,dt

for

:$$Re(a)>0wedgeRe(s)>0wedge z<1veeRe(a)>0wedgeRe(s)>1wedge z=1.

A contour integral representation is given by

:$$Phi(z,s,a)=-frac{Gamma(1-s)}{2pi i}int_{0}^{(+infty)}frac{(-t)^{s-1}e^{-at{1-ze^{-t,dt

for

:$$Re(a)>0wedgeRe(s)<0wedge z<1

where the contour must not enclose any of the points $$t=log(z)+2kpi i,kin Z.

A Hermite-like integral representation is given by

:$$Phi(z,s,a)=frac{1}{2a^s}+int_{0}^{infty}frac{z^{t{(a+t)^{s,dt+frac{2}{a^{s-1int_{0}^{infty}frac{sin(sarctan(t)-talog(z))}{(1+t^2)^{s/2}(e^{2pi at}-1)},dt

for

:$$Re(a)>0wedge |z|<1

and

:$$Phi(z,s,a)=frac{1}{2a^s}+frac{log^{s-1}(1/z)}{z^a}Gamma(1-s,alog(1/z))+frac{2}{a^{s-1int_{0}^{infty}frac{sin(sarctan(t)-talog(z))}{(1+t^2)^{s/2}(e^{2pi at}-1)},dt

for :$$Re(a)>0.

pecial cases

The Hurwitz zeta-function is a special case, given by:$,$zeta(s,alpha)=L(0, alpha,s)=Phi(1,s,alpha).

The polylogarithm is a special case of the Lerch Zeta, given by :$,$extrm{Li}_s(z)=zPhi(z,s,1).

The Legendre chi function is a special case, given by:$,$chi_n(z)=2^{-n}z Phi (z^2,n,1/2).

The Riemann zeta-function is given by:$,$zeta(s)=Phi (1,s,1).

The Dirichlet eta-function is given by:$,$eta(s)=Phi (-1,s,1).

Identities

For &lambda; rational, the summand is a root of unity, and thus $$L(lambda, alpha, s) may be expressed as a finite sum over the Hurwitz zeta-function.

Various identities include::$$Phi(z,s,a)=z^n Phi(z,s,a+n) + sum_{k=0}^{n-1} frac {z^k}{(k+a)^s}

and

:$$Phi(z,s-1,a)=left(a+zfrac{partial}{partial z} ight) Phi(z,s,a)

and

:$$Phi(z,s+1,a)=-,frac{1}{s}frac{partial}{partial a} Phi(z,s,a).

eries representations

A series representation for the Lerch transcendent is given by

:$$Phi(z,s,q)=frac{1}{1-z} sum_{n=0}^infty left(frac{-z}{1-z} ight)^nsum_{k=0}^n (-1)^k {n choose k} (q+k)^{-s}.

The series is valid for all "s", and for complex "z" with Re("z")&lt;1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.

A Taylor's series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for :$$|log(z)|<2 pi;s eq 1,2,3,dots; a eq 0,-1,-2,dots:$$Phi(z,s,a)=z^{-a}left [Gamma(1-s)left(-log (z) ight)^{s-1}+sum_{k=0}^{infty}zeta(s-k,a)frac{log^{k}(z)}{k!} ight]

:"(the correctness of this formula is disputed, please see the )"Please see:B. R. Johnson,Generalized Lerch zeta-function.Pacific J. Math. 53, no. 1 (1974), 189–193."http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.pjm/1102911791?abstract= "

If s is a positive integer, then:$$Phi(z,n,a)=z^{-a}left{sum_k=0}atop k eq n-1}^{infty}zeta(n-k,a)frac{log^{k}(z)}{k!}+left [Psi(n)-Psi(a)-log(-log(z)) ight] frac{log^{n-1}(z)}{(n-1)!} ight}.

A Taylor series in the third variable is given by:$$Phi(z,s,a+x)=sum_{k=0}^{infty}Phi(z,s+k,a)(s)_{k}frac{(-x)^k}{k!};|x|

Series at "a" = -"n" is given by:$$Phi(z,s,a)=sum_{k=0}^{n}frac{z^k}{(a+k)^s}+z^nsum_{m=0}^{infty}(1-m-s)_{m}Li_{s+m}(z)frac{(a+n)^m}{m!}; a ightarrow-n A special case for "n" = 0 has the following series:$$Phi(z,s,a)=frac{1}{a^s}+sum_{m=0}^{infty}(1-m-s)_{m}Li_{s+m}(z)frac{a^{m{m!}; |a|<1.

An asymptotic series for $$s ightarrow-infty:$$Phi(z,s,a)=z^{-a}Gamma(1-s)sum_{k=-infty}^{infty} [2kpi i-log(z)] ^{s-1}e^{2kpi ai}for $$|a|<1;Re(s)<0 ;z otin (-infty,0) and:$$Phi(-z,s,a)=z^{-a}Gamma(1-s)sum_{k=-infty}^{infty} [(2k+1)pi i-log(z)] ^{s-1}e^{(2k+1)pi ai}for $$|a|<1;Re(s)<0 ;z otin (0,infty).

An asymptotic series in the incomplete Gamma function:$$Phi(z,s,a)=frac{1}{2a^s}+frac{1}{z^a}sum_{k=1}^{infty}frac{e^{-2pi i(k-1)a}Gamma(1-s,a(-2pi i(k-1)-log(z)))} {(-2pi i(k-1)-log(z))^{1-s+frac{e^{2pi ika}Gamma(1-s,a(2pi ik-log(z)))}{(2pi ik-log(z))^{1-sfor $$|a|<1;Re(s)<0.

References

* Mathias Lerch, "Démonstration élémentaire de la formule: $$frac{pi^2}{sin^2{pi x=sum_{ u=-infty}^{infty}frac{1}{(x+ u)^2}", (1903), L'Enseignement Mathématique, 5, pp.450-453.
* M. Jackson, "On Lerch's transcendent and the basic bilateral hypergeometric series $$,_2psi_2", (1950) J. London Math. Soc., 25, pp. 189-196
* H. Bateman, "Higher Transcendental Functions", (1953) McGraw-Hill, New York.
* A. Erdélyi, "Higher Transcendental Functions", (1953) McGraw-Hill, New York.
* Ramunas Garunkstis, " [http://www.mif.vu.lt/~garunkstis Home Page] " (2005) "(Provides numerous references and preprints.)"
* Jesus Guillera and Jonathan Sondow, " [http://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent] " (2005) "(Includes various basic identities in the introduction.)"
* Ramunas Garunkstis, " [http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf Approximation of the Lerch Zeta Function] " (PDF)
* Sergej V. Aksenov and Ulrich D. Jentschura, " [http://aksenov.freeshell.org/lerchphi.html C and Mathematica Programs for Calculation of Lerch's Transcendent] " (2002)

* S. Kanemitsu, Y. Tanigawa and H. Tsukada, " [http://www.iisc.ernet.in/nias/HRJ/vol27/Ktt.pdf A generalization of Bochner's formula] ", (undated, 2005 or earlier)

* A. Laurinv cikas,R. Garunkv stis, "The Lerch zeta-function.", Kluwer Academic Publishers, Dordrecht, 2002. viii+189 pp.